However, there are many interesting classes of infinite dimensional
Lie groups which possess a rich unitary representation theory.
Many of these representation show up naturally in various contexts
of mathematical physics ([1]},
[2]}, [3]}, [4]},
[5]}, [6]}, [7]}, [8]},
[9]}).
The representations arising in mathematical physics, resp.,
Quantum Field Theory are often characterized by the
requirement that the Lie algebra \({\mathfrak {g}}= \mathop {\bf L{}}\nolimits (G)\) 
of \(G\)  contains an element \(h\) , corresponding
to the Hamiltonian of the underlying physical system, for which the
spectrum of the operator \(i\cdot {\tt d}\pi (h)\) 
in the “physically relevant” representations \((\pi , \mathcal {H})\) 
is non-negative. These representations are called positive energy
representations (cf. [10]}, [11]}, [5]}, [13]}).
